The energy requirement to move a car, even at low speed, is substantial. The average person is lucky to put out 200 watts of power using his muscular strength on a bicycle setup. Even a pro cyclist puts out only half a kilowatt (500 watts). An extremely underpowered car has a power plant of several 10's of kilowatts (the electric power plant of the Prius is about 50 kW; it is coupled with a 50 hp gas engine). Even an electric golf cart has a powerplant which puts out 2 to 3 kilowatts of power.

Have you ever tried to push a car at slow speed up even a very small slope? Regardless of how the mechanism is geared, this is how much power you'd have to apply. You can gear the mechanism down, to trade rotational speed for torque, but the speed you'd move the car will be well under 3 kph, especially uphill, even those uphills we consider "false flats" which look flat but are really a very shallow up hill.

To make this work, you'd have to limit the weight of the car to something like 100, maybe 200 lbs on the outside, the weight of a fully loaded touring bike. You'd have to use your legs, not your arms, as your leg muscles are much stronger than any muscle in the upper body. You'd might as well just use a touring bike to bike, and forget the engine, or use one of those electric assists.

Cat 2 Track, Cat 3 Road.
"If you’re new enough [to racing] that you would ask such question, then i would hazard a guess that if you just made up a workout that sounded hard to do, and did it, you’d probably get faster." --the tiniest sprinter

I should add that even a 50 hp gas engine (which is extremely small for a car - most cars have power plants of over 100 hp), required for traveling freeway speeds, is well over 100 lbs in weight.

Cat 2 Track, Cat 3 Road.
"If you’re new enough [to racing] that you would ask such question, then i would hazard a guess that if you just made up a workout that sounded hard to do, and did it, you’d probably get faster." --the tiniest sprinter

Brian says that it will not work, but I am not convinced. Sometimes one just need to move a car a meter or two. On aircrafts, on bombers during the 2nd War, one could control the landing geer manually.

I can push my car easily, I would like to have some foldable levers, so I could move it if the engine is down.

RE bombers during WWII: I don't remember the movie, and of course, it was just a movie, but they showed two crew members cranking on that handle for a long time to get the gear down and locked. And, of course, gravity is assisting to lower the landing gear.

Like I said though, you can do it; you can move anything given enough leverage. You just won't move it very fast. Work is force times distance. You can decrease the force towards zero as long as you don't mind the distance going to infinity.

Cat 2 Track, Cat 3 Road.
"If you’re new enough [to racing] that you would ask such question, then i would hazard a guess that if you just made up a workout that sounded hard to do, and did it, you’d probably get faster." --the tiniest sprinter

The idea of a low power backup for low speed is good - things like nearly stopped bumper to bumper traffic, parking lots, moving car in driveway, etc.

However the added complexity is probably not worth the benefit.

The direction the technology is going to to store excess power electrically and use this energy in part for low speed travel needs. I think given this trend, that using human power for this backup is even less likely to be practical - especially as this battery power can store more total and give more peak energy than the average human is capable of.

suppose that we got the task to implement it on the conceptual level. Where to start?

Technically the challenge to add a pedal power input system to a car is not that difficult - made somewhat more difficult due to the need to integrate into existing parts vs. from a ground up design which would be easier to design but harder to prototype - So the proof of concept here is not so interesting a challenge.

The real challenges this would face is the human interface. Here is where you would need to put thought into it - how does one add pedal power input from drivers seat while also providing and not limiting all normal auto controls.

The work around is to put pedal system in any but drivers seat - but that misses the majority of auto use.

Figure out how you want the interface to integrate into a typical car first.

I'm pretty sure that this exists - albeit a very lightweight car. Someone posted it a while back. It was supposed to go around 50mph IIRC on battery and pedaling. You could just pedal if you wanted. Around $20,000 for something that was more bicycle than car, though.

If you redesigned the cockpit, make the accelerator and brake controls hand operated leaving your feet to do the pedalling. You'll need to think about the steering interface as a normal steering wheel setup does not leave much room for the motion of pedalling with ones legs. Seat design would be critical as well. Perhaps you'd want to adopt a recument type position in the driver's seat. My guess is that after you've thought out all of the major issues and designed the vehicle to be as light as possible such that your pedal power yields a reasonable power to weight ratio, you'll end up with a bicycle with a small gas engine on it.